Exercise 2.5.3

Question

\begin{enumerate}[label=(\alph*)] 
  \item Prove that if an infinite series converges, then the associative property holds. Assume $a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots$ converges to a limit $L$ (i.e., the sequence of partial sums $\left.\left(s_{n}\right) \rightarrow L\right)$. Show that any regrouping of the terms
    $$
    \left(a_{1}+a_{2}+\cdots+a_{n_{1}}\right)+\left(a_{n_{1}+1}+\cdots+a_{n_{2}}\right)+\left(a_{n_{2}+1}+\cdots+a_{n_{3}}\right)+\cdots
    $$
    leads to a series that also converges to $L$.
  \item Compare this result to the example discussed at the end of Section $2.1$ where infinite addition was shown not to be associative. Why doesn't our proof in (a) apply to this example?
   \end{enumerate}

Ulisse Mini · Accepted Answer

\begin{enumerate}[label=(\alph*)] 
  \item Let $s_n$ be the original partial sums, and let $s_m'$ be the regrouping.
    Since $s_m'$ is a subsequence of $s_n$, $(s_n) 	o s$ implies $(s_m') 	o s$.
  \item The subsequence $s_m' = (1 - 1) + \dots = 0$ converging does not imply the parent sequence $s_n$ converges.
    In fact BW tells us any bounded sequence of partial sums will have a convergent subsequence (regrouping in this case).
   \end{enumerate}

Exercise 2.5.3

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Exercise 2.5.3

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